Kinematics of Mechanisms from the Time of Watt Part 2

[Ill.u.s.tration: Figure 12.--Cartwright's geared straight-line mechanism of about 1800. From Abraham Rees, _The Cyclopaedia_ (London, 1819, "Steam Engine," pl. 5).]

The properties of a hypocycloid were recognized by James White, an English engineer, in his geared design which employed a pivot located on the pitch circle of a spur gear revolving inside an internal gear. The diameter of the pitch circle of the spur gear was one-half that of the internal gear, with the result that the pivot, to which the piston rod was connected, traced out a diameter of the large pitch circle (fig.

13). White in 1801 received from Napoleon Bonaparte a medal for this invention when it was exhibited at an industrial exposition in Paris.[29] Some steam engines employing White's mechanism were built, but without conspicuous commercial success. White himself rather agreed that while his invention was "allowed to possess curious properties, and to be a _pretty_ thing, opinions do not all concur in declaring it, essentially and generally, a _good_ thing."[30]

[Footnote 29: H. W. d.i.c.kinson, "James White and His 'New Century of Inventions,'" _Transactions of the Newcomen Society_, 1949-1951, vol.

27, pp. 175-179.]

[Footnote 30: James White, _A New Century of Inventions_, Manchester, 1822, pp. 30-31, 338. A hypocycloidal engine used in Stourbridge, England, is in the Henry Ford Museum.]

[Ill.u.s.tration: Figure 13.--James White's hypocycloidal straight-line mechanism, about 1800. The fly-weights (at the ends of the diagonal arm) functioned as a flywheel. From James White, _A New Century of Inventions_ (Manchester, 1822, pl. 7).]

The first of the non-Watt four-bar linkages appeared shortly after 1800.

The origin of the gra.s.shopper beam motion is somewhat obscure, although it came to be a.s.sociated with the name of Oliver Evans, the American pioneer in the employment of high-pressure steam. A similar idea, employing an isosceles linkage, was patented in 1803 by William Freemantle, an English watchmaker (fig. 14).[31] This is the linkage that was attributed much later to John Scott Russell (1808-1882), the prominent naval architect.[32] An inconclusive hint that Evans had devised his straight-line linkage by 1805 appeared in a plate ill.u.s.trating his _Abortion of the Young Steam Engineer's Guide_ (Philadelphia, 1805), and it was certainly used on his Columbian engine (fig. 15), which was built before 1813. The Freemantle linkage, in modified form, appeared in Rees's _Cyclopaedia_ of 1819 (fig. 16), but it is doubtful whether even this would have been readily recognized as identical with the Evans linkage, because the connecting rod was at the opposite end of the working beam from the piston rod, in accordance with established usage, while in the Evans linkage the crank and connecting rod were at the same end of the beam. It is possible that Evans got his idea from an earlier English periodical, but concrete evidence is lacking.

[Footnote 31: British Patent 2741, November 17, 1803.]

[Footnote 32: William J. M. Rankine, _Manual of Machinery and Millwork_, ed. 6, London, 1887, p. 275.]

[Ill.u.s.tration: Figure 14.--Freemantle straight-line linkage, later called the Scott Russell linkage. From British Patent 2741, November 17, 1803.]

[Ill.u.s.tration: Figure 15.--Oliver Evans' "Columbian" engine, 1813, showing the Evans, or "gra.s.shopper," straight-line linkage. From _Emporium of Arts and Sciences_ (new ser., vol. 2, no. 3, April 1814, pl. opposite p. 380).]

[Ill.u.s.tration: Figure 16.--Modified Freemantle linkage, 1819, which is kinematically the same as the Evans linkage. Pivots _D_ and _E_ are attached to engine frame. From Abraham Rees, _The Cyclopaedia_ (London, 1819, "Parallel Motions," pl. 3).]

If the idea did in fact originate with Evans, it is strange that he did not mention it in his patent claims, or in the descriptions that he published of his engines.[33] The practical advantage of the Evans linkage, utilizing as it could a much lighter working beam than the Watt or Freemantle engines, would not escape Oliver Evans, and he was not a man of excessive modesty where his own inventions were concerned.

[Footnote 33: Greville and Dorothy Bathe, _Oliver Evans_, Philadelphia, 1935, pp. 88, 196, and _pa.s.sim_.]

Another four-bar straight-line linkage that became well known was attributed to Richard Roberts of Manchester (1789-1864), who around 1820 had built one of the first metal planing machines, which machines helped make the quest for straight-line linkages largely academic. I have not discovered what occasioned the introduction of the Roberts linkage, but it dated from before 1841. Although Roberts patented many complex textile machines, an inspection of all of his patent drawings has failed to provide proof that he was the inventor of the Roberts linkage.[34]

The fact that the same linkage is shown in an engraving of 1769 (fig.

18) further confuses the issue.[35]

[Footnote 34: Robert Willis (_op. cit._ [Footnote 21] p. 411) credited Richard Roberts with the linkage. Roberts' 15 British patent drawings exhibit complex applications of cams, levers, guided rods, cords, and so forth, but no straight-line mechanism. In his patent no. 6258 of April 13, 1832, for a steam engine and locomotive carriage, Roberts used Watt's "parallel motion" on a beam driven by a vertical cylinder.]

[Footnote 35: This engraving appeared as plate 11 in Pierre Patte's 1769 work (_op. cit._ footnote 24). Patte stated that the machine depicted in his plate 11 was invented by M. de Voglie and was actually used in 1756.]

[Ill.u.s.tration: Figure 17.--Straight-line linkage (before 1841) attributed to Richard Roberts by Robert Willis. From A. B. Kempe, _How to Draw a Straight Line_ (London, 1877, p. 10).]

[Ill.u.s.tration: Figure 18.--Machine for sawing off pilings under water, about 1760, designed by De Voglie. The Roberts linkage operates the bar (_Q_ in detailed sketch) at the rear of the machine below the operators.

The significance of the linkage apparently was not generally recognized.

A similar machine depicted in Diderot's _Encyclopedie_, published several years later, did not employ the straight-line linkage. From Pierre Patte, _Memoirs sur les objets plus importants de l'architecture_ (Paris, 1769, pl. 11).]

The appearance in 1864 of Peaucellier's exact straight-line linkage went nearly unnoticed. A decade later, when news of its invention crossed the Channel to England, this linkage excited a flurry of interest, and variations of it occupied mathematical minds for several years. For at least 10 years before and 20 years after the final solution of the problem, Professor Chebyshev,[36] a noted mathematician of the University of St. Petersburg, was interested in the matter. Judging by his published works and his reputation abroad, Chebyshev's interest amounted to an obsession.

[Footnote 36: This is the Library of Congress spelling]

Pafnut[)i] L'vovich Chebyshev was born in 1821, near Moscow, and entered the University of Moscow in 1837. In 1853, after visiting France and England and observing carefully the progress of applied mechanics in those countries, he read his first paper on approximate straight-line linkages, and over the next 30 years he attacked the problem with new vigor at least a dozen times. He found that the two princ.i.p.al straight-line linkages then in use were Watt's and Evans'. Chebyshev noted the departure of these linkages from a straight line and calculated the deviation as of the fifth degree, or about 0.0008 inch per inch of beam length. He proposed a modification of the Watt linkage to refine its accuracy but found that he would have to more than double the length of the working beam. Chebyshev concluded ruefully that his modification would "present great practical difficulties."[37]

[Footnote 37: _Oeuvres de P. L. Tchebychef_, 2 vols., St. Petersburg, 1899-1907, vol. 1, p. 538; vol. 2, pp. 57, 85.]

At length an idea occurred to Chebyshev that would enable him to approach if not quite attain a true straight line. If one mechanism was good, he reasoned, two would be better, _et cetera, ad infinitum_. The idea was simply to combine, or compound, four-link approximate linkages, arranging them in such a way that the errors would be successively reduced. Contemplating first a combination of the Watt and Evans linkages (fig. 19), Chebyshev recognized that if point D of the Watt linkage followed nearly a straight line, point A of the Evans linkage would depart even less from a straight line. He calculated the deviation in this case as of the 11th degree. He then replaced Watt's linkage by one that is usually called the Chebyshev straight-line mechanism (fig.

20), with the result that precision was increased to the 13th degree.[38] The steam engine that he displayed at the Vienna Exhibition in 1873 employed this linkage--the Chebyshev mechanism compounded with the Evans, or approximate isosceles, linkage. An English visitor to the exhibition commented that "the motion is of little or no practical use, for we can scarcely imagine circ.u.mstances under which it would be more advantageous to use such a complicated system of levers, with so many joints to be lubricated and so many pins to wear, than a solid guide of some kind; but at the same time the arrangement is very ingenious and in this respect reflects great credit on its designer."[39]

[Footnote 38: _Ibid._, vol. 2, pp. 93, 94.]

[Footnote 39: _Engineering_, October 3, 1873, vol. 16, p. 284.]

[Ill.u.s.tration: Figure 19.--Pafnut[)i] L'vovich Chebyshev (1821-1894), Russian mathematician active in a.n.a.lysis and synthesis of straight-line mechanisms. From _Ouvres de P. L. Tchebychef_ (St. Petersburg, 1907, vol. 2, frontispiece).]

[Ill.u.s.tration: Figure 20.--Chebyshev's combination (about 1867) of Watt's and Evans' linkages to reduce errors inherent in each. Points _C_, _C'_, and _C"_ are fixed; _A_ is the tracing point. From _Oeuvres de P. L. Tchebychef_ (St. Petersburg, 1907, vol. 2, p. 93).]

[Ill.u.s.tration: Figure 21.--_Top_: Chebyshev straight-line linkage, 1867; from A. B. Kempe, _How to Draw a Straight Line_ (London, 1877, p. 11).

_Bottom_: Chebyshev-Evans combination, 1867; from _Oeuvres de P. L.

Tchebychef_ (St. Petersburg, 1907, vol. 2, p. 94). Points _C_, _C'_, and _C"_ are fixed. _A_ is the tracing point.]

There is a persistent rumor that Professor Chebyshev sought to demonstrate the impossibility of constructing any linkage, regardless of the number of links, that would generate a straight line; but I have found only a dubious statement in the _Grande Encyclopedie_[40] of the late 19th century and a report of a conversation with the Russian by an Englishman, James Sylvester, to the effect that Chebyshev had "succeeded in proving the nonexistence of a five-bar link-work capable of producing a perfect parallel motion...."[41] Regardless of what tradition may have to say about what Chebyshev said, it is of course well known that Captain Peaucellier was the man who finally synthesized the exact straight-line mechanism that bears his name.

[Footnote 40: _La Grande Encyclopedie_, Paris, 1886 ("Peaucellier").]

[Footnote 41: James Sylvester, "Recent Discoveries in Mechanical Conversion of Motion," _Notices of the Proceedings of the Royal Inst.i.tution of Great Britain_, 1873-1875, vol. 7, p. 181. The fixed link was not counted by Sylvester; in modern parlance this would be a six-link mechanism.]

[Ill.u.s.tration: Figure 22.--Peaucellier exact straight-line linkage, 1873. From A. B. Kempe, _How to Draw a Straight Line_ (London, 1877, p.

12).]

[Ill.u.s.tration: Figure 23.--Model of the Peaucellier "Compas Compose,"

deposited in Conservatoire National des Arts et Metiers, Paris, 1875.

Photo courtesy of the Conservatoire.] [Ill.u.s.tration: Figure 24.--James Joseph Sylvester (1814-1897), mathematician and lecturer on straight-line linkages. From _Proceedings of the Royal Society of London_ (1898, vol. 63, opposite p. 161).]

Charles-Nicolas Peaucellier, a graduate of the Ecole Polytechnique and a captain in the French corps of engineers, was 32 years old in 1864 when he wrote a short letter to the editor of _Nouvelles Annales de mathematiques_ (ser. 2, vol. 3, pp. 414-415) in Paris. He called attention to what he termed "compound compa.s.ses," a cla.s.s of linkages that included Watt's parallel motion, the pantograph, and the polar planimeter. He proposed to design linkages to describe a straight line, a circle of any radius no matter how large, and conic sections, and he indicated in his letter that he had arrived at a solution.

This letter stirred no pens in reply, and during the next 10 years the problem merely led to the filling of a few academic pages by Peaucellier and Amedee Mannheim (1831-1906), also a graduate of Ecole Polytechnique, a professor of mathematics, and the designer of the Mannheim slide rule.

Finally, in 1873, Captain Peaucellier gave his solution to the readers of the _Nouvelles Annales_. His reasoning, which has a distinct flavor of discovery by hindsight, was that since a linkage generates a curve that can be expressed algebraically, it must follow that any algebraic curve can be generated by a suitable linkage--it was only necessary to find the suitable linkage. He then gave a neat geometric proof, suggested by Mannheim, for his straight-line "compound compa.s.s."[42]

[Footnote 42: Charles-Nicholas Peaucellier, "Note sur une question de geometrie de compas," _Nouvelles Annales de mathematiques_, 1873, ser.

2, vol. 12, pp. 71-78. A sketch of Mannheim's work is in Florian Cajori, _A History of the Logarithmic Slide Rule_, New York, about 1910, reprinted in _String Figures and Other Monographs_, New York, Chelsea Publis.h.i.+ng Company, 1960.]

On a Friday evening in January 1874 Albemarle Street in London was filled with carriages, each maneuvering to unload its charge of gentlemen and their ladies at the door of the venerable hall of the Royal Inst.i.tution. Amidst a "mighty rustling of silks," the elegant crowd made its way to the auditorium for one of the famous weekly lectures. The speaker on this occasion was James Joseph Sylvester, a small intense man with an enormous head, sometime professor of mathematics at the University of Virginia, in America, and more recently at the Royal Military Academy in Woolwich. He spoke from the same rostrum that had been occupied by Davy, Faraday, Tyndall, Maxwell, and many other notable scientists. Professor Sylvester's subject was "Recent Discoveries in Mechanical Conversion of Motion."[43]

[Footnote 43: Sylvester, _op. cit._ (footnote 41), pp. 179-198. It appears from a comment in this lecture that Sylvester was responsible for the word "linkage." According to Sylvester, a linkage consists of an even number of links, a "link-work" of an odd number. Since the fixed member was not considered as a link by Sylvester, this distinction became utterly confusing when Reuleaux's work was published in 1876.

Although "link" was used by Watt in a patent specification, it is not probable that he ever used the term "link-work"--at any rate, my search for his use of it has been fruitless. "Link work" is used by Willis (_op. cit._ footnote 21), but the term most likely did not originate with him. I have not found the word "linkage" used earlier than Sylvester.]

Remarking upon the popular appeal of most of the lectures, a contemporary observer noted that while many listeners might prefer to hear Professor Tyndall expound on the acoustic opacity of the atmosphere, "those of a higher and drier turn of mind experience ineffable delight when Professor Sylvester holds forth on the conversion of circular into parallel motion."[44]

[Footnote 44: Bernard H. Becker, _Scientific London_, London, 1874, pp.

45, 50, 51.]

Sylvester's aim was to bring the Peaucellier linkage to the notice of the English-speaking world, as it had been brought to his attention by Chebyshev--during a recent visit of the Russian to England--and to give his listeners some insight into the vastness of the field that he saw opened by the discovery of the French soldier.[45]